We prove a dichotomy: in any class of tournaments \(\mathcal {T}\)  (closed under subtournaments), FOMC is either \(\textrm {FPT}\)  or \(\textrm {AW}[*]\) -hard on \(\mathcal {T}\) .
More precisely, we show that the key of this dichotomy is twin-width,
a complexity parameter introduced by Bonnet et al. [1]}, defined on arbitrary binary structures.
If \(\mathcal {T}\)  has bounded twin-width, then FOMC in \(\mathcal {T}\)  is \(\textrm {FPT}\) ,
whereas it is \(\textrm {AW}[*]\) -hard when \(\mathcal {T}\)  has unbounded twin-width.
This dichotomy also coincides with a model theoretic characterisation:
the class \(\mathcal {T}\)  has bounded twin-width if and only if it is (monadically) NIP,
which roughly speaking means that it is not possible
to describe all graphs from \(\mathcal {T}\)  using a fixed first-order formula.
In the vocabulary of [2]}, this proves that tournaments are delineated:
every subclass is monadically NIP if and only if it has bounded twin-width,
answering a question posed in [2]}.
The equivalence between NIP and \(\textrm {FPT}\)  FO model checking also
confirms the `nowhere FO dense' conjecture of Gajarský et al. [4]} for classes of tournaments.
